10.2. Slope Ratio Method

This is a general purpose procedure that can be used to
analyse balanced or unbalanced assays with blanks (0-dose treatments), plate
(row) effects and unlimited numbers of dose levels and test preparations. The
algorithm is based on Finney (1978). The Slope Ratio Method specification given in European Pharmacopoeia (1997-2008), is a restricted special case of this procedure.

10.2.1.
Slope Ratio Variable Selection

The data format is as in Parallel Line Method (see 10.1.1. Data Preparation).
Measurement data is stacked in a single column, a second column contains the
dose level for each measurement and another categorical column indicates which
preparation a particular measurement belongs to. An optional row factor can be
entered to keep track of the replicates.

Designs can be unbalanced, i.e. the number of replicates for
each dose-preparation combination may be different, dose levels for standard
and test preparations may be different, there can be more than one test
preparation, but the first preparation should always be the standard. It is
compulsory to select at least three columns [Data], [Dose] and
[Preparation]. The optional [Row Factor] column is usually
used to isolate a plate effect (the replicates) and when one is selected, the
program assumes that all dose/treatment groups (or cells) have an equal number
of replicates.

10.2.2.2. Homogeneity of Variance Tests

10.2.2.3. Validity of Assay

This output option displays an Analysis of Variance (ANOVA) table, which is used in testing the Validity of Assay. The standard significance tests performed are (i)
regression, (ii) intercept and (iii) non-linearity. The overall non-linearity
test is also broken down to individual tests for each preparation. If blanks
(entries with a 0 dose level) exist, there will be an additional term for them.
If a [Row Factor] was selected it will appear in the table as a
main effect.

Define a cell as a unique combination of dose levels
and preparations. For each cell calculate:

Define the overall mean as:

where N is the total number of observations. Also define and as
the intercept and slope for each preparation from Separate
Regression and as the slope for
each preparation from Common Regression (see 10.2.2.4. Regression).

The following definitions are used in calculating the blanks
effect:

where Sxxi is as defined in Separate Regression and:

Also define the number of unique dose-preparation
combinations excluding blanks as:

The ANOVA table is then constructed as follows.

Due to

Degrees of Freedom

Sum of Squares

Plate

K – 1

SSP

Between Doses

D – 1 + B

SSD

Blanks

B = 0 or 1

SSB

Regression

n

SSR

Intercept

n – 1

SSD – SSB

– SSR – SSL

Non-linearity

D – 2n

SSL

Non-linearity for

Preparationi

D / n – 2

SSLi

Residual

N – D – B – (K – 1)

SSE – SSP

–
SSP

Total

N – 1

SST

10.2.2.4. Regression

Calculate for i = S, T1, …, Tn – 1:

The estimated parameters of the line of best fit for each
preparation (i = S, T1, …, Tn – 1) are:

Slope:

R-squared:

Residual sum of squares:

Standard error of slope:

This information is displayed in the Separate
Regression table and used in drawing the best fit lines in Plot of Treatments.

The Common Regression is
obtained from a multivariate regression run, after transforming the data into
the following form first.

Dependent

Independent Variables

Variable

Standard

Test 1

Test n

Blank

Y0jk

0

0

0

Replicates

…

…

…

…

Standard

YSjk

XSjk

0

0

Replicates

…

…

…

…

Test 1

Y1jk

0

X1jk

0

Replicates

…

…

…

…

Test n

Ynjk

0

0

Xnjk

Replicates

…

…

…

…

The estimated parameters are
displayed in Common Regression table.

10.2.2.5. Potency

By default, each test preparation is assigned a potency of unity. If you want to change this click the [Opt] button situated to the left of
the Potency option. In this case, a further
dialogue pops up asking for entry of assigned potency for each test preparation.

For each test preparation, the potency ratio is calculated as follows:

For confidence intervals of M first define Vss, Vii, Vsi, i
= T1, …, Tn – 1 as the values corresponding to elements
of (X’X)-1 matrix from the Common Regression
run. First define:

where s2 is the residual mean squares and is the critical value from the
t-distribution with degrees of freedom of the overall residual term from the
ANOVA table. Note that if divided by s2, Vss, Vii, Vsi give the
variance / covariance matrix of the Common Regression
coefficients.

Then the confidence interval for potency ratio of each test preparation is calculated using Fieller’s Theorem (see Finney 1978, p. 156):

where the variance of Mi is:

and the approximate variance of Mi is (when g is
negligible):

Mi is the relative potency and MiL and MiU are the confidence limits for the relative
potency. The estimated potency and its confidence interval are obtained by
multiplying these relative values by the assigned potency supplied by the user for each test preparation separately.

Weights are computed after the estimated potency and its confidence interval are found:

and % Precision is:

10.2.2.6. Plot of
Treatments

This option generates a Plot of Treatments against dose levels. Standard and each test
preparation are plotted in separate series and a line of best fit is drawn for
each one of them. The coefficients of lines are as in Separate
Regression output.

Clicking the [Opt] button situated to the left of the Plot of Treatments option will place the graph in
UNISTAT’s Graphics Editor. The plot can be further customised and annotated
using the tools available under the UNISTAT Graphics Window’s Edit menu.

The same plot is drawn here using the X-Y Plots procedure, this time with confidence intervals for
regression lines included.

10.2.3.
Slope Ratio Examples

Example 1

Data is given in Table 5.2.1-I on p. 588 of European
Pharmacopoeia (2008). The data is rearranged as described in section 10.1.1. Data Preparation
and saved in columns 24-26 of BIOPHARMA6.

Although the data set contains blanks (0 dose treatments),
they need to be removed from the analysis. In Excel Add-In Mode, you can simply select the block X10:Z57. In Stand-Alone Mode, you can define C26 as a Select Row column to omit these rows from the analysis,
without actually deleting them from the spreadsheet. To do this, click
somewhere on column 26, and select Data → Select Row option from UNISTAT’s spreadsheet menus. The colour
of C26 will change. This indicates that all rows with a 0 entry in this
column will be omitted from subsequent analyses.

Select Bioassay → Slope Ratio Method. In Stand-Alone Mode select columns C23, C24 and L25 respectively
as [Data], [Dose] and [Preparation] from the Variable Selection
Dialogue. In Excel Add-In Mode, you will need to select the three highlighted
columns in the same order. Click [Next] to proceed to Output Options Dialogue. If you do not want to display all normality tests
click on the [Opt] button situated to the left of Normality
Tests option. Click [None] and then check the Shapiro-Wilk
Test and Report summary statistics boxes.
Then click [Back] and [Finish].

In Stand-Alone Mode, do not forget to reset column 4 after you finish
this example, otherwise the Select Row function will be effective in subsequent procedures
you run. To do this, click somewhere on column 4, and select Data → Select Row option again, or select Formula → Quick Formula from the menu and enter data. The colour of C26
will change back to its original value.

Slope Ratio Method

Rows 1-8 Omitted

Selected by C26 Select

Normality Tests

Smaller probabilities indicate non-normality.

* Lilliefors probability = 0.2 means 0.2 or greater.

DosexPreparations

Valid Cases

Mean

Standard Deviation

Shapiro-Wilk Test

Probability

1 x Standard S

8

0.1351

0.0025

0.8969

0.2707

2 x Standard S

8

0.2176

0.0021

0.8816

0.1952

3 x Standard S

8

0.2996

0.0027

0.8269

0.0551

1 x Preparation T

8

0.1200

0.0011

0.8599

0.1199

2 x Preparation T

8

0.1898

0.0012

0.8042

0.0318

3 x Preparation T

8

0.2554

0.0018

0.9255

0.4763

Homogeneity of Variance Tests

Test Statistic

Probability

Bartlett’s Chi-square Test

8.5820

0.1269

Bartlett-Box F Test

1.7315

0.1239

Cochran’s C (max var / sum var)

0.3079

0.3345

Hartley’s F (max var / min var)

6.2344

0.0500

p > 0.05

Levene’s F Test

2.2830

0.0635

Validity of Assay

Due To

Sum of
Squares

DoF

Mean Square

F-Stat

Prob

Constant

1.976

1

1.976

Regression

0.192

2

0.096

24849.565

0.0000

Intercept

0.000

1

0.000

0.001

0.9780

Non-linearity

0.000

2

0.000

2.984

0.0614

Standard S
Non-linearity

0.000

1

0.000

0.086

0.7702

Preparation T
Non-linearity

0.000

1

0.000

5.882

0.0197

Treatments

0.192

5

0.038

Residual

0.000

42

0.000

Total

0.192

47

0.004

Separate Regression

Intercept

Slope

Residual SS

R-squared

Standard S

0.0530

0.0822

0.0001

0.9989

Preparation T

0.0530

0.0677

0.0001

0.9992

Common Regression

Intercept

Slope

Residual SS

R-squared

Standard S

0.0530

0.0822

0.0002

0.9990

Preparation T

0.0677

Potency

Test Preparation

Assigned Potency

Estimated Potency

Lower 95%

Upper 95%

Preparation T

1.0000

0.8231

0.8171

0.8292

Test Preparation

Variance

Weight

% Precision

Preparation T

0.0000

110860.3771

99.2644

G =

0.0001

C =

1.0001

Example 2

Data is given in Table 5.2.2-I on p. 589 of European
Pharmacopoeia (2008).

Open BIOPHARMA6 and select Bioassay
→ Slope Ratio Method. The blank preparation is already omitted from this
data set. From the Variable Selection
Dialogue select columns C27, C28
and L29 Preparations respectively as [Data], [Dose] and
[Preparation]. Click [Next] to proceed to Output Options Dialogue. Click on the [Opt] button situated to the left of Normality Tests option, click [None] and then check the Cramer-von Mises Test and Report
summary statistics boxes and click [Back]. Click the [Opt] button
situated to the left of the Potency option. Enter
the assigned potency value 15 for both preparations, click [Back] and [Finish].

Slope Ratio Method

Normality Tests

Smaller probabilities
indicate non-normality.

DosexPreparations

Valid Cases

Mean

Standard Deviation

Cramer-von Mises Test

Probability

1 x Standard S

2

18.0000

0.0000

*

*

2 x Standard S

2

23.6500

1.2021

0.0419

0.4774

3 x Standard S

2

30.4000

0.0000

*

*

4 x Standard S

2

36.1500

0.6364

0.0419

0.4774

1 x Preparation T

2

15.9500

1.2021

0.0419

0.4774

2 x Preparation T

2

23.6500

0.7778

0.0419

0.4774

3 x Preparation T

2

28.1500

1.0607

0.0419

0.4774

4 x Preparation T

2

36.1000

2.4042

0.0419

0.4774

1 x Preparation U

2

15.5500

0.2121

0.0419

0.4774

2 x Preparation U

2

19.4000

1.1314

0.0419

0.4774

3 x Preparation U

2

23.6500

0.7778

0.0419

0.4774

4 x Preparation U

2

27.2000

0.2828

0.0419

0.4774

Homogeneity of Variance Tests

Test Statistic

Probability

Bartlett’s Chi-square Test

5.2396

0.8129

Bartlett-Box F Test

0.5751

0.8146

Cochran’s C (max var / sum var)

0.4510

0.2364

Hartley’s F (max var / min var)

128.4444

Levene’s F Test

Validity of Assay

Due To

Sum of Squares

DoF

Mean Square

F-Stat

Prob

Constant

14785.770

1

14785.770

Regression

1087.665

3

362.555

339.498

0.0000

Intercept

3.474

2

1.737

1.626

0.2371

Non-linearity

5.065

6

0.844

0.791

0.5943

Standard S Non-linearity

0.446

2

0.223

0.209

0.8144

Preparation T Non-linearity

4.453

2

2.227

2.085

0.1670

Preparation U Non-linearity

0.166

2

0.083

0.078

0.9257

Treatments

1096.205

11

99.655

Residual

12.815

12

1.068

Total

1109.020

23

48.218

Separate Regression

Intercept

Slope

Residual SS

R-squared

Standard S

11.7500

6.1200

2.2960

0.9939

Preparation T

9.7250

6.4950

13.4085

0.9692

Preparation U

11.6500

3.9200

2.1760

0.9860

Common Regression

Intercept

Slope

Residual SS

R-squared

Standard S

11.0417

6.3561

21.3544

0.9807

Preparation T

6.0561

Preparation U

4.1228

Potency

Test Preparation

Assigned Potency

Estimated Potency

Lower 95%

Upper 95%

Preparation T

15.0000

14.2920

13.3681

15.2711

Preparation U

15.0000

9.7295

8.8542

10.6088

Test Preparation

Variance

Weight

% Precision

Preparation T

0.0008

5.2437

93.5355

Preparation U

0.0007

6.1678

91.0034

G =

0.0056

C =

1.0056

Example 3

Table 7.10.2. on p. 161 from Finney, D. J. (1978) is an example with blanks, four replicates and two preparations.

Open BIOFINNEY and select Bioassay
→ Slope Ratio Method. From the Variable Selection
Dialogue select columns C12 Data,
C13 Dose and S14 Preparations respectively as [Data], [Dose]
and [Preparation]. Click [Next] to proceed to Output Options Dialogue. Click [All] to select all output options and then
click [Finish]. The potency ratio and its confidence limits are calculated with
the default assigned potency of 1. The following output is obtained.